Limits in Presheaf Categories

Summary

Limits (and colimits) in functor categories $[\mathcal{A},\mathcal{B}]$ are computed pointwise when $\mathcal{B}$ has the relevant limits. Applying this to $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$: all limits and colimits exist and are pointwise. The Density Theorem says every presheaf is a colimit of representables, with the colimit shaped by the category of elements.

Overview

Presheaf categories inherit excellent limit/colimit properties from Set. The pointwise computation makes checking limits easy. The density theorem, in contrast, shows that the representables “generate” the entire presheaf category via colimits — a deep structural fact.

Main Content

Pointwise Limits in Functor Categories

Theorem 6.2.5: Limits in Functor Categories are Pointwise (BCT, Ch. 6.2)

Let $\mathcal{B}$ have all limits of shape $\mathcal{I}$, and let $\mathcal{D}:\mathcal{J}\to [\mathcal{I},\mathcal{B}]$ be a diagram of functors. Then:

1. $[\mathcal{I},\mathcal{B}]$ has limits of shape $\mathcal{J}$.
2. The limit $\lim \mathcal{D}$ is computed pointwise: $(\lim \mathcal{D}{)}_I={\lim }_J{\mathcal{D}}_J(I)$ for each $I\in \mathcal{I}$.

Consequence for presheaf categories: Since Set is complete and cocomplete, $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ is complete and cocomplete, with all limits and colimits computed pointwise:

$$(\lim X_j)(A)=\underset{j}{\lim }{X}_j(A),(\mathrm{colim}{X}_j)(A)={\mathrm{colim}}_j{X}_j(A)$$

Example: Product of Presheaves (BCT, Ch. 6.2)

For presheaves $X,Y:{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$, their product is $(X\times Y)(A)=X(A)\times Y(A)$.

Limits Commute with Limits

Proposition 6.2.8: Limits Commute (BCT, Ch. 6.2)

For a diagram $D:\mathcal{I}\times \mathcal{J}\to \mathcal{A}$ (when all relevant limits exist):

$$\underset{\mathcal{I}\times \mathcal{J}}{\lim }D\cong \underset{I}{\lim }\underset{J}{\lim }D(I,J)\cong \underset{J}{\lim }\underset{I}{\lim }D(I,J)$$

This generalises the commutativity of products and intersections in set theory.

Yoneda Embedding Preserves Limits

Corollary 6.2.12: Yoneda Preserves Limits (BCT, Ch. 6.2)

The Yoneda embedding $H^{\bullet }:\mathcal{A}\to [{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ preserves all small limits.

Proof: By Proposition 6.2.2, each $H_A=\mathcal{A}(-,A)$ preserves limits (as a right adjoint in the representable sense). Since the limit in $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ is pointwise, $H^{\bullet }(\lim D)(A)=\mathcal{A}(A,\lim D)\cong \lim \mathcal{A}(A,D(-))=\lim H^{\bullet }(D)(A)$.

Category of Elements

Definition 6.2.16: Category of Elements

For a functor $X:{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$, the category of elements $\mathcal{E}(X)$ (also written $\int X$ or ${\int }^{A}X(A)$) has:

• Objects: pairs $(A,x)$ where $A\in \mathcal{A}$ and $x\in X(A)$
• Morphisms $(A,x)\to (B,y)$: maps $f:A\to B$ in $\mathcal{A}$ such that $(Xf)(y)=x$

There is a forgetful functor $P:\mathcal{E}(X)\to \mathcal{A}$, $(A,x)\mapsto A$.

Density Theorem

Theorem 6.2.17: Density Theorem (BCT, Ch. 6.2)

For any small $\mathcal{A}$ and any presheaf $X\in [{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$:

$$X\cong {\mathrm{colim}}_{(A,x)\in \mathcal{E}(X)}{H}_A$$

i.e., $X$ is the colimit of the diagram $\mathcal{E}(X)\stackrel{P}{\to }\mathcal{A}\stackrel{H^{\bullet }}{\to }[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$.

Interpretation: Every presheaf is “built from” representable presheaves via colimits. The category of elements $\mathcal{E}(X)$ encodes exactly how to assemble $X$ from representables.

Analogy: A set is a colimit of one-element sets; a presheaf is a colimit of representables (which are the “one-element sets” of the presheaf world).

Example: Density for Representables

When $X=H_B$, the category of elements has objects $(A,f:A\to B)$ — i.e., maps into $B$. The density colimit is $H_B\cong {\mathrm{colim}}_{f:A\to B}{H}_A$, which is a Yoneda-type statement.

Connections

• The cartesian closed structure of $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ (Cartesian Closed Categories) also relies on limits being pointwise.
• The density theorem is used in the proof of the General Adjoint Functor Theorem (Adjoint Functor Theorems).

See Also

• Functor Categories — Presheaf categories
• General Limits — Limits in general
• Yoneda Embedding and Consequences — Yoneda embedding preserves limits
• Cartesian Closed Categories — Presheaf categories are cartesian closed